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I know that all Banach spaces are (Hausdorff)locally convex spaces. I would like to verify that the converse is not true by giving an example of a space which is locally convex but not Banach. I am looking at the linear space $C([0,1],\left\|\cdot\right\|_1)$ of complex-valued functions defined on the compact interval $[0,1]$, where the norm is given by $$\left\|f\right\|_1=\int_{0}^{1}|f(t)|dt $$ for any $f\in C[0,1]$. The normed space $C([0,1],\left\|\cdot\right\|_1)$ is known to be not Banach. I don't have any idea if the said space is a locally convex space. Any tips? Thanks in advance...

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    $\begingroup$ Every normed space is Hausdorff and locally convex. A Banach space is a complete normed space. If you are looking at normed spaces, to be "non-Banach" is the same as to be incomplete. $\endgroup$ – Willie Wong Feb 1 '13 at 14:01
  • $\begingroup$ Maybe you can add the definition of local convexity you are trying to verify and add some thoughts (or a proof) why Willie's first sentence is correct. $\endgroup$ – Martin Feb 1 '13 at 14:18
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This is sasisfied by the Schawrtz space, which is locally convex but isn't Banach. There is a class of spaces called Frechet spaces that are locally convex and not necessarily Banach. Wikipedia has a few examples.

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